To determine the x-values that make the function f(x)=(4x²+5)(x-6)²(x+8)³ equal to zero, we can benefit from the zero product property. This property states that if a product of factors is equal to zero, then at least one of the factors must be zero.
Our equation is already factored, which simplifies things. The factors are (4x²+5), (x-6)², and (x+8)³.
We set each of these factors equal to zero and solve for x.
Step 1: Set the factor 4x²+5 equal to zero and solve for x.
4x² + 5 = 0
4x² = -5
x² = -5/4
In this case, the value under the square root is negative, which means that the square root is not a real number, it's a complex number. Therefore, there are no real solutions for x that come from this factor.
Step 2: Set the factor (x-6)² equal to zero and solve for x.
(x - 6)² = 0
x - 6 = 0
x = 6
So, there is one real solution that comes from this factor, which is x = 6.
Step 3: Set the factor (x+8)³ equal to zero and solve for x.
(x + 8)³ = 0
x + 8 = 0
x = -8
So, there is another real solution that comes from this factor, which is x = -8.
In conclusion, the real x-values that cause the function to be zero are x = 6 and x = -8. There are no other real solutions.